The Gutenberg-Richter law states that the size-frequency distribution of earthquakes follows a power law. This trend is usually justified using spring-block models, where slips with the appropriate statistics of sizes have been numerically observed. However, prominent spatial and temporal clustering features of earthquakes, as those implied by the Omori law of aftershocks, are not accounted for by this kind of model unless they are complemented with ad hoc assumptions, such as stress recovery laws after slip events, or the phenomenological rate-and-state equations to describe friction. We show that when a mechanism of structural relaxation is incorporated into a spring-block model, realistic earthquake patterns following the Gutenberg-Richter and Omori laws are obtained. Moreover, features well known from laboratory friction experiments, such as velocity weakening and increase of static friction with contact time, appear as a consequence of the relaxational mechanism as well, without making any a priori assumptions on the velocity dependence of the friction force in the model. In this way, our model shows that a single physical mechanism may be a unifying concept behind the Gutenberg-Richter and Omori laws and the rate-and-state equations of rock friction.